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Limit properties for ratios of order statistics from exponentials
- Yong Zhang^{1}Email authorView ORCID ID profile and
- Xue Ding^{1}
https://doi.org/10.1186/s13660-016-1287-6
© The Author(s) 2017
- Received: 4 July 2016
- Accepted: 23 December 2016
- Published: 6 January 2017
Abstract
In this paper, we study the limit properties of the ratio for order statistics based on samples from an exponential distribution and obtain the expression of the density functions, the existence of the moments, the strong law of large numbers for \(R_{nij}\) with \(1\leq i< j< m_{n}=m\). We also discuss other limit theorems such as the central limit theorem, the law of iterated logarithm, the moderate deviation principle, the almost sure central limit theorem for self-normalized sums of \(R_{nij}\) with \(2\leq i< j< m_{n}=m\).
Keywords
- exponential distribution
- order statistics
- strong law of large numbers
- central limit theorem
- law of iterated logarithm
1 Introduction and main results
Theorem A
Later on, Miao et al. [2] proved the central limit theorem and the almost sure central limit for \(R_{n23}\) with fixed sample size, we state their results as the following theorem.
Theorem B
In this paper, we will make a further study on the limit properties of \(R_{nij}\). In the next section, firstly, we give the expression of the density functions of \(R_{nij}\) for all \(1\leq i< j< m_{n}\), it is more interesting that the density function is free of the sample mean \(\lambda_{n}\), this allows us to change the equipment from sample to sample as long as the underlying distribution remains an exponential. Also we discuss the existence of the moments for fixed sample size \(m_{n}=m\). Secondly, we establish the strong law of large number for \(R_{nij}\) with \(1=i< j< m\) and \(2\leq i< j< m\), respectively. At last we give some limit theorems such as the central limit theorem, the law of iterated logarithm, the moderate deviation principle, the almost sure central limit theorem for self-normalized sums of \(R_{nij}\) with \(2\leq i< j< m\).
In the following, C denotes a positive constant, which may take different values whenever it appears in different expressions. \(a_{n}\sim b_{n}\) means that \(a_{n}/b_{n}\rightarrow1\) as \(n\rightarrow \infty\).
2 Main results and proofs
2.1 Density functions and moments of \(R_{nij}\)
The first theorem gives the expression of the density functions.
Theorem 2.1
Proof
The next theorem treats the moments of \(R_{nij}\) with fixed sample size \(m_{n}=m\).
Theorem 2.2
Proof
For \(2\leq i< j\leq m\), similarly we can obtain \(f_{nij}(r)\sim\frac{d_{m,i,j}}{r^{3}}\), where \(d_{m,i,j}\) is a constant depend only on m, i and j, so the γ-order moment is finite for \(0<\gamma<2\) and is infinite for \(\gamma\geq2\). Furthermore it is not difficult to verify that \(L_{1}(r)=E R_{nij}^{2}I\{|R_{nij}|\leq r\}\) varies slowly at ∞, then by the fact that if \(L(x)=E|X|^{2}I\{|X|\leq x\}\) is a slowly varying function at ∞, then \(L_{a}(x)=E|X-a|^{2}I\{|X-a|\leq x\}\) also varies slowly at ∞ for any \(a\in R\), the proof is completed. □
Remark 2.3
Miao et al. [2] obtained the density function for \(R_{n2j}\) for fixed sample size \(m_{n}=m\), they also proved that the expectation of \(R_{n2j}\) is finite and the truncated second moment is slowly varying at ∞. Adler [1] also claimed that all the \(R_{n1j}\) have infinite expectations for fixed sample size, so our theorems extended their results.
2.2 Strong law of large numbers of \(R_{nij}\)
From our assumptions, we know that \(\{R_{nij},n\geq1\}\) is an independent sequence with the same distribution for fixed sample size \(m_{n}=m\). As Theorem 2.2 states that the \(R_{n1j}\) do not have the expectation, so the strong law of large numbers with them is not typical. Here we give the weighted strong law of large number as follows. At first, we list the following lemma, that is, Theorem 2.6 from De la Peña et al. [3], which will be used in the proof.
Lemma 2.4
Let \(\{X_{n},n\geq1\}\) be a sequence of independent random variables, denote \(S_{n}=\sum_{i=1}^{n}X_{i}\), if \(b_{n}\nearrow\infty\), and \(\sum_{i=1}^{\infty}\operatorname{Var}(X_{i})/{b_{i}^{2}}<\infty\), then \((S_{n}-ES_{n})/{b_{n}}\to 0\) a.s.
Theorem 2.5
Proof
By the same argument as in the proof of (2.4), we can get (2.5), so we omit it here. □
Remark 2.6
If we take \(a_{n}=\frac{(\log n)^{\alpha}}{n}\), \(b_{n}=(\log n)^{\alpha +2}\), \(\alpha>-2\), it is easy to check that conditions (2.2) and (2.3) hold with \(\lambda=\frac{1}{\alpha+2}\), so Theorems 2.1 and 2.2 and 4.1 from Adler [1] are special cases of our Theorem 2.5. There are some other sequences satisfying conditions (2.2) and (2.3), such as (a) \(a_{n}=1\), \(b_{n}=n^{\beta}\), \(\beta>1\), \(\lambda=0\); (b) \(a_{n}=1\), \(b_{n}=n(\log n)^{\gamma}\), \(\gamma>1\), \(\lambda=0\); (c) \(a_{n}=1\), \(b_{n}=n(\log n)(\log\log n)^{\delta}\), \(\delta>1\), \(\lambda =0\); (d) \(a_{n}=\frac{(\log\log n)^{\theta}}{n}\), \(b_{n}=(\log n)^{2}(\log \log n)^{\theta}\), \(\theta\in R\), \(\lambda=\frac{1}{2}\), so the conditions (2.2) and (2.3) are mild conditions. At the end of this remark, we point out that only when \(a_{n}=L(n)/n\), where \(L(n)\) is a slowly varying function, the limit value λ will be \(\lambda>0\), this is known as an exact strong law, one can refer to Adler [4] for more details. For the weak law, i.e., convergence in probability, one can see Feller [5] for full details.
For \(R_{nij}\), \(i\geq2\), since the expectation is finite, by the classical strong law of large numbers, we have the following.
Theorem 2.7
2.3 Other limit properties for \(R_{nij}\), \(2\leq i< j\leq m\)
By the above discussion, we know that, for fixed sample size \(m_{n}=m\) and \(2\leq i< j\leq m\), \(\{R_{nij},n\geq1\}\) is a sequence of independent and identically distributed random variables with finite mean, and \(L(r)=E(R_{nij}-ER_{nij})^{2}I\{|R_{nij}-ER_{nij}|\leq r\}\) is a slowly varying function at ∞. Therefore the limit properties of \(R_{nij}\) for fixed sample size can easily be established by those of the self-normalized sums. We list some of them, such as the central limit theorem (CLT), the law of iterated logarithm (LIL), the moderate deviation principle (MDP), the almost sure central limit theorem (ASCLT). Denote \(S_{N}=\sum_{n=1}^{N}(R_{nij}-ER_{nij})\), \(V_{N}^{2}=\sum_{n=1}^{N}(R_{nij}-ER_{nij})^{2}\).
Theorem 2.8
CLT
Proof
By Theorem 3.3 from Giné et al. [6], we can obtain the CLT for \(R_{nij}\). □
Theorem 2.9
LIL
Proof
By Theorem 1 from Griffin and Kuelbs [7], the LIL for \(R_{nij}\) holds. □
Theorem 2.10
MDP
Proof
By Theorem 3.1 from Shao [8], we can prove the MDP for \(R_{nij}\). □
Theorem 2.11
ASCLT
Proof
By Corollary 1 from Zhang [9], we know ASCLT for \(R_{nij}\) holds. □
Declarations
Acknowledgements
This work was supported by National Natural Science Foundation of China (Grant Nos. 11101180, 11201175); the Science and Technology Development Program of Jilin Province (Grant Nos. 20130522096JH, 20140520056JH).
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Authors’ Affiliations
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